Algebraic space

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In mathematics, an algebraic space is a generalization of the schemes of algebraic geometry introduced by Michael Artin for use in deformation theory.

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[edit] Definition

An algebraic space X comprises a scheme[1] U and a closed subscheme R ⊂ U × U satisfying the following two conditions:

1. R is an equivalence relation as a subset of U × U
2. The projections pi: R → U onto each factor are étale maps.

If a third condition

3. R is the trivial equivalence relation over each connected component of U

is satisfied, then the algebraic space will be a scheme in the usual sense. Thus, an algebraic space allows a single connected component of U to cover X with many "sheets". The point set underlying the algebraic space X is then given by |U| / |R| as a set of equivalence classes.

Let Y be an algebraic space defined by an equivalence relation S ⊂ V × V. The set Hom(Y, X) of morphisms of algebraic spaces is then defined by the condition that it makes the descent sequence

\mathrm{Hom}(Y, X) \rightarrow \mathrm{Hom}(V, X) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \mathrm{Hom}(S, X)

exact (this definition is motivated by a descent theorem of Grothendieck for surjective étale maps of affine schemes). With these definitions, the algebraic spaces form a category.

Let U be an affine scheme over a field k defined by a system of polynomials g(x), x = (x1, …, xn), let

k{x1, …, xn}

denote the ring of algebraic functions in x over k, and let X = {R ⊂ U × U} be an algebraic space.

The appropriate stalks ÕX, x on X are then defined to be the local rings of algebraic functions defined by ÕU, u, where u ∈ U is a point lying over x and ÕU, u is the local ring corresponding to u of the ring

k{x1, …, xn} / (g)

of algebraic functions on U.

A point on an algebraic space is said to be smooth if ÕX, x ≅ k{z1, …, zd} for some indeterminates z1, …, zd. The dimension of X at x is then just defined to be d.

A morphism f: Y → X of algebraic spaces is said to be étale at y ∈ Y (where x = f(y)) if the induced map on stalks

ÕX, x → ÕY, y

is an isomorphism.

The structure sheaf OX on the algebraic space X is defined by associating the ring of functions O(V) on V (defined by étale maps from V to the affine line A1 in the sense just defined) to any algebraic space V which is étale over X.

[edit] Facts about algebraic spaces

  • Algebraic curves are schemes.
  • Non-singular algebraic surfaces are schemes.
  • Algebraic spaces with group structure are schemes.
  • Not every singular algebraic surface is a scheme.
  • Not every non-singular 3-dimensional algebraic space is a scheme.
  • Every algebraic space contains a dense open affine subscheme, and the complement of such a subscheme always has codimension ≥ 1. Thus algebraic spaces are in a sense "close" to affine schemes.

[edit] Applications

To be written

[edit] See also

[edit] Notes

  1. ^ One can always assume that U is an affine scheme. Doing so means that the theory of algebraic spaces is not dependent on the full theory of schemes, and can indeed be used as a (more general) replacement of that theory.

[edit] References

  • Artin, Michael. Algebraic Spaces. Yale University Press, 1971.
  • Knutson, Donald. Algebraic Spaces. Springer Lecture Notes in Mathematics, 203, 1971.